Properties

Label 2-528-1.1-c1-0-9
Degree $2$
Conductor $528$
Sign $-1$
Analytic cond. $4.21610$
Root an. cond. $2.05331$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 4·7-s + 9-s − 11-s − 2·13-s − 2·15-s − 2·17-s − 4·21-s − 8·23-s − 25-s + 27-s − 6·29-s + 8·31-s − 33-s + 8·35-s + 6·37-s − 2·39-s − 2·41-s − 2·45-s − 8·47-s + 9·49-s − 2·51-s + 6·53-s + 2·55-s + 4·59-s + 6·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.485·17-s − 0.872·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s + 1.35·35-s + 0.986·37-s − 0.320·39-s − 0.312·41-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.269·55-s + 0.520·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-1$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.05246880525653227640097293961, −9.723139098205724899141727122835, −8.548353092044889979587894449296, −7.75720782023225106841673661541, −6.87928263143909248442940323738, −5.90723738552130668805725208182, −4.35461089374668409702872860477, −3.53074178558790394542885743604, −2.45250023112523203194083612695, 0, 2.45250023112523203194083612695, 3.53074178558790394542885743604, 4.35461089374668409702872860477, 5.90723738552130668805725208182, 6.87928263143909248442940323738, 7.75720782023225106841673661541, 8.548353092044889979587894449296, 9.723139098205724899141727122835, 10.05246880525653227640097293961

Graph of the $Z$-function along the critical line